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Every non-negative binary number base 2 has a corresponding decimal base 10 expansion form. This can be written in decimal as a sum of products: As you can see, for every binary digit that is 1 we will add a power of 2 to the decimal representation. The power of 2 to be added corresponds to the placement of the binary digit indexed from 0, starting on the right. If a binary digit is 0, then it will zero out its corresponding power of two, contributing nothing to the decimal expansion.
If you consider each product in the sum to be a component corresponding to a bit, then:. Check out the following site if you're unfamiliar with how to convert decimal numbers to their binary representation: The first method is analogous to the standard algorithm. The second method is one that may be quicker if performing the conversion by hand e. Let's assume you have an int called, "result," and you have a way of determining what a bit in its binary representation looks like.
In order to toggle a binary bit, you need to first know if the bit is 0 or 1. To figure this out, we can utilize shifting which will covered in more detail below. This works because the bit we want to check will end up being the first bit on the right after the shift. If the bit is 1, then the shifted number will be odd. If the bit is 0, then the shifted number will be even. This should make sense of you examine the decimal expansion of the shifted number.
Here is an example implementation:. If a bit in the binary representation gets shifted off the edge in either direction, it's lost. Shifting to the left shifts the bits to the left:. This can be seen in the examples above, but can also be proved by examining the decimal expansion.
This can be written as: The binary expansion of this shifted number can be written as: It therefore follows that: My name is Michael E. The best way to make sense of michaelcotterell. You can also subscribe via RSS. This website was designed by Michael E.
Cotterell using Twitter Bootstrap. The header image is provided under a Creative Commons license by by Jonathan Hinkle. The content and opinions expressed on this Web site do not necessarily reflect the views of nor are they endorsed by the University of Georgia or the University System of Georgia. Toggle navigation Michael E. Blog Contact Vitae Research Publications. This article was originally written to help my CSCI students out on a project. Binary to Decimal Every non-negative binary number base 2 has a corresponding decimal base 10 expansion form.
If you consider each product in the sum to be a component corresponding to a bit, then: The the left hand side of the product represents the value of the bit 0 or 1.
The right hand side of the product the power of 2 represents the digit placement of the bit. Toggling Bits Let's assume you have an int called, "result," and you have a way of determining what a bit in its binary representation looks like. Hopefully this is obvious based on how the decimal expansion, explained earlier, works. Here is an example implementation: Shifting to the left shifts the bits to the left: Shifting to the right shifts the bits to the right: